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Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. (English) Zbl 1250.35134
The model consists of a wave equation on a bounded domain, coupled with a pointwise damped harmonic oscillator equation, with boundary conditions on the interface. The purpose of the paper is to study the existence, uniqueness and uniform decay of finite energy solutions to the considered model. In case some specific conditions are fulfilled, the existence and uniqueness of weak solutions to the system, continuously depending on initial data, are proved. Under additional assumptions, decay rates for the energy functional for solutions of the PDE system are obtained. Techniques from nonlinear semigroup theory are used. For similar investigations by the author, see [the author and B. Said-Hourari, J. Differ. Equations 252, No. 9, 4898–4941 (2012; Zbl 1250.35135)].

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs
Full Text: DOI
[1] Alber H., Cooper J.: Quasilinear hyperbolic 2 x 2 systems with a free, damping boundary condition. Journal für die reine und angewandte Mathematik 406, 10–43 (1990) · Zbl 0702.35149
[2] Avalos G.: The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis 1, 203–217 (1996) · Zbl 0989.93078 · doi:10.1155/S1085337596000103
[3] Avalos G., Lasiecka I.: The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system. Semigroup Forum 57, 278–292 (1998) · Zbl 0910.35014 · doi:10.1007/PL00005977
[4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, 1976. · Zbl 0328.47035
[5] Beale J. T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25, 895–917 (1976) · Zbl 0332.35050 · doi:10.1512/iumj.1976.25.25071
[6] Beale J. T.: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26, 199–222 (1977) · Zbl 0352.35071 · doi:10.1512/iumj.1977.26.26015
[7] Beale J. T., Rosencrans S. I.: Acoustic boundary conditions. Bull. Amer. Math. Soc. 80, 1276–1278 (1974) · Zbl 0294.35045 · doi:10.1090/S0002-9904-1974-13714-6
[8] Cousin A. T., Frota C. L., Larkin N. A.: Global solvability and asymptotic behaviour of hyperbolic problem with acoustic boundary conditions. Funkcial. Ekvac. 44, 471–485 (2001) · Zbl 1145.35433
[9] Cousin A. T., Frota C. L., Larkin N. A.: On a system of klein-gordon type equations with acoustic boundary conditions. Journal of Mathematical Analysis and Applications 293, 293–309 (2004) · Zbl 1060.35118 · doi:10.1016/j.jmaa.2004.01.007
[10] Frota C. L., Larkin N. A.: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Progress in Nonlinear Differential Equations and Their Applications 66, 297–312 (2005) · Zbl 1105.35018 · doi:10.1007/3-7643-7401-2_20
[11] Gal C., Goldstein G., Goldstein J.: Oscillatory boundary conditions for acoustic wave equations. Journal of Evolution Equations 3, 623–635 (2003) · Zbl 1058.35139 · doi:10.1007/s00028-003-0113-z
[12] Graber P.: Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system. Nonlinear Analysis: Theory and Applications 73, 3058–3068 (2010) · Zbl 1200.35193 · doi:10.1016/j.na.2010.06.075
[13] Graber P.: Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions. Nonlinear Analysis: Theory and Applications 74, 3058–3068 (2011) · Zbl 1200.35193
[14] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, vol. 50 of Progress in Nonlinear Differential Equations, 2002, pp. 197–217. · Zbl 1027.35003
[15] Lasiecka I.: Mathematical Control Theory of Coupled PDEs. SIAM, Philadelphia (2002) · Zbl 1032.93002
[16] Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations 6, 507–533 (1993) · Zbl 0803.35088
[17] Lasiecka I., Triggiani R.: Uniform stabilization of the wave equation with dirichlet or neumann feedback control without geometrical conditions. Applied Mathematics and Optimization 25, 189–224 (1992) · Zbl 0780.93082 · doi:10.1007/BF01182480
[18] Morse P. M., Ingard K. U.: Theoretical Acoustics. McGraw-Hill, New York (1968)
[19] J. Y. Park and T. G. Ha, Well-posedness for the klein gordon equation with damping term and acoustic boundary conditions, J. Math. Physics, 50 (2009). · Zbl 1200.35190
[20] Sell G., You Y.: Dynamics of Evolutionary Equations. Springer, New York (2002) · Zbl 1254.37002
[21] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, vol. 49 of Mathematical Surveys and Monographs, AMS, 1997. · Zbl 0870.35004
[22] Temam R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin-Heidelberg-New York (1988) · Zbl 0662.35001
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